The Goldbach conjecture with summands in arithmetic progressions

Abstract

We prove that, for almost all $r\le N^{1/2}/{\log }^{O(1)}N$, for any given $b_1\mathrm{mod}r$ with $(b_1,r)=1$, and for almost all $b_2\mathrm{mod}r$ with $(b_2,r)=1$, we have that almost all natural numbers $2n\le N$ with $2n\equiv b_1+b_2\mathrm{mod}r$ can be written as the sum of two prime numbers $2n=p_1+p_2$, where $p_1\equiv b_1\mathrm{mod}r$ and $p_2\equiv b_2\mathrm{mod}r$. This improves the previous result which required $r\le N^{1/3}/{\log }^{O(1)}N$ instead of $r\le N^{1/2}/{\log }^{O(1)}N$. We also improve some other results concerning variations of the problem.